On the connection formulas of the third Painlevé transcendent

Roderick S. C. Wong; H. Y. Zhang

doi:10.3934/dcds.2009.23.541

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2009, Volume 23, Issue 1&2: 541-560. Doi: 10.3934/dcds.2009.23.541

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On the connection formulas of the third Painlevé transcendent

Roderick S. C. Wong^1, and
H. Y. Zhang^2,

1.
Department of Mathematics, City University of Hong Kong, Kowloon
2.
Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong

Received: November 2007

Revised: February 2008

Published: May 2009

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Abstract

We consider the connection problem for the sine-Gordon PIII equation$u_{x x}+\frac{1}{x}u_{x}+\sin u=0,$ which is the most commonly studied case among all general thirdPainlevé transcendents. The connectionformulas are derived by the method of "uniform asymptotics"proposed by Bassom, Clarkson, Law and McLeod (Arch. Rat. Mech.Anal., 1998).

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Mathematics Subject Classification: Primary: 33E17, 34M55; Secondary: 35Q53.

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